Population genetics

Foundations

Per Unneberg

NBIS

12/15/22

Population genetics

DNA variation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T

The main data for molecular population genetics are DNA sequences. The alignment above shows a sample of four DNA sequences. Each sequence has 15 nucleotides (sites) “from the same locus (location) on a chromosome” (p.2 Hahn, 2019)

Alternative names for sequence:

  • chromosome
  • gene
  • allele (different by origin)
  • sample
  • cistron

We will preferentially use sequence or chromosome to refer to an entire sequence, and allele to refer to individual nucleotides that differ.

DNA variation - monomorphic sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T
* * * * * * * * * T

The alignment has 4 DNA sequences where each sequence has length \(L=15\). A site where all nucleotides (alleles) are identical is called a monomorphic site (indicated with asterisks above). There are 9 monomorphic sites.

DNA variation - segregating sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T
* * * * * *

A site where there are different nucleotides (alleles) is called a segregating site (indicated with asterisks above), often denoted S. There are \(S=6\) segregating sites.

Alternative names for segregating site are:

  • polymorphism
  • mutation
  • single nucleotide polymorphism (SNP)

mutation may refer to the process that generates new variation and the new variants generated by this process, which is the way we will use the term.

In contrast to mutation which corresponds to within-species variation, a substitution refers to DNA differences between species.

DNA variation - major and minor alleles

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T
* * * * * *

Much of the nucleotide variation we study consists of bi-allelic SNPs. The most common variant is called the major allele, and the least common the minor allele.

The set of alleles found on a single sequence is called haplotype.

Describing DNA variation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T

Once we have a sample of sequences we want to describe the observed variation. At any position the ith allele has sample frequency \(p_i\), where the sum of all allele frequencies is 1. For instance, at site 1, \(p_T=1\) (and by extension \(p_A=p_C=p_G=0\)), and at site 2 \(p_C=1/4\) and \(p_T=3/4\).

Heterozygosity

The heterozygosity at a site is given by

\[ h = \frac{n}{n-1}\left(1 - \sum p_i^2\right) \]

Exercise: calculate the heterozygosity at sites 1 and 2

\[ h_1 = \frac{4}{3} \left(1 - p_T^2 \right) = 0 \\ h_2 = \frac{4}{3} \left(1 - \left(p_C^2 + p_T^2\right) \right) = \frac{4}{3} \left( 1 - \left(\frac{1}{16} + \frac{9}{16}\right)\right) = \frac{1}{2} \]

Describing DNA variation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T T A C A A T C C G A T C G T
T T A C G A T G C G C T C G T
T C A C A A T G C G A T G G A
T T A C G A T G C G C T C G T

Nucleotide diversity \(\pi\)

The nucleotide diversity is the sum of site heterozygosities:

\[ \pi = \sum_{j=1}^S h_j \]

Calculate the nucleotide diversity

Observation: \(h_i\) either 1/2 or 2/3 (for sites with \(p_{major}=p_{minor}\)).

\[ \pi = \frac{1}{2} + \frac{2}{3} + \frac{1}{2} + \frac{2}{3} + \frac{1}{2} + \frac{1}{2} = 3\frac{1}{3} \]

Often we provide \(\pi\) per site:

\[ \pi = 3.33/15 = 0.222 \]

Origin and change of variation

Wright-Fisher model

Model of evolution that describes sampling of alleles in a population under the following assumptions:

  • no selection
  • no mutation
  • no migration
  • non-overlapping generations
  • random mating

Algorithm

  1. Setup starting population at time zero

Wright-Fisher model

Model of evolution that describes sampling of alleles in a population under the following assumptions:

  • no selection
  • no mutation
  • no migration
  • non-overlapping generations
  • random mating

Algorithm

  1. Setup starting population at time zero

Wright-Fisher model

Model of evolution that describes sampling of alleles in a population under the following assumptions:

  • no selection
  • no mutation
  • no migration
  • non-overlapping generations
  • random mating

Algorithm

  1. Setup starting population at time zero
  2. Add offspring (same size) at time one

Wright-Fisher model

Model of evolution that describes sampling of alleles in a population under the following assumptions:

  • no selection
  • no mutation
  • no migration
  • non-overlapping generations
  • random mating

Algorithm

  1. Setup starting population at time zero
  2. Add offspring (same size) at time one
  3. Select parents to offspring at random

Wright-Fisher model

Model of evolution that describes sampling of alleles in a population under the following assumptions:

  • no selection
  • no mutation
  • no migration
  • non-overlapping generations
  • random mating

Algorithm

  1. Setup starting population at time zero
  2. Add offspring (same size) at time one
  3. Select parents to offspring at random

Wright-Fisher model

Introduce model as way to understand drift. Show n=10 and interactively/iteratively build next generation(s). Forward simulation. (Add pgip code)

Demonstrate this is binomial sampling.

Wright-Fisher model

Genetic drift

To capture dynamics, follow allele frequency trajectory as function of time.

Code 
wright_fisher <- function(p0, N, generations) {
    x <- vector(mode="numeric", length=generations)
    x[1] <- p0
    for (i in seq(2, length(x)))
        x[i] <- rbinom(1, size=N, prob=x[i-1]) / N
    x
}

Figure 1: Genetic drift for different combinations of starting frequency and population size

Genetic drift

Figure 2: Genetic drift for different combinations of starting frequency and population size for n=50 repetitions per parameter combination. Note how variation and time to fixation depends on population size and starting frequency.

Mutation

Drift randomly “moves” frequencies, but once lost, how does new variation arise? Mutation.

Introduce variation at rate \(2N_e\mu\)

Mutation - drift balance

Infinite sites vs infinite alleles

Selection

Relative fitness.

Define s, positive and negative selection

Direct selection

d_N/d_S

Two-locus dynamics

Graph focal locus and flanking loci (Charlesworth figure)

  • Linkage disequilibrium and recombination (each probably need separate treatment)
  • Contrast linked selection with direct selection
  • Genetic draft / background selection

Neutral theory

Move right after drift?

  • Hahn: \(\mu = \nu*f_0\)
  • why population size matters
  • Fig 4.2, Eq 4.2 from Lynch

Nearly neutral theory

  • Distribution of fitness effects?

A primer on simulation

If simulation section show brief code examples

Wright-Fisher model again

Repeat WF model. Point out that it can be used for forward simulation. Show how many individuals are lost each generation (limit e-1): lose lots of compute resources on the way to extant sample.

Solution: look backwards

The coalescent

Introduce concept. Describe algorithm (show code - include as exercise?). Genealogy and mutations:

  • sim_ancestry
  • sim_mutations

Note that #mutations propto branch length -> don’t actually even need to know the mutations (e.g. fastsimcoal)

The coalescent: diminishing returns

Show plot of diminishing returns (better to add sites, not samples)

The coalescent and diversity

Recalculate example

Evolutionary processes and genealogies

Non-neutral processes change topologies in ways that we detect when applying tests (Ferretti et al., 2017)

genealogies change due to non Show some qualitative examples of how trees change for

  • bottleneck

msprime

SLiM

Mention advances in forward simulations

Recapitation

Combine the best of two worlds

Recipes

Show some recipes and the pgip CLI

Population genetics foundations

Ferretti, L., Ledda, A., Wiehe, T., Achaz, G., & Ramos-Onsins, S. E. (2017). Decomposing the Site Frequency Spectrum: The Impact of Tree Topology on Neutrality Tests. Genetics, 207(1), 229–240. https://doi.org/10.1534/genetics.116.188763
Hahn, M. (2019). Molecular Population Genetics (First). Oxford University Press.
Hein, J., Schierup, M., & Wiuf, C. (2004). Gene genealogies, variation and evolution. A primer in coalescent theory. In Systematic Biology - SYST BIOL (Vol. 54).